Alexander Lubotzky Hebrew University Let M=M(g) be the mapping class group of a surface of genus g>1 (resp., M=Aut(Fg), the automorphism group of the free group on g generators). As it is well known, M is mapped onto the symplectic group Sp(2g,Z) (resp., the general linear group GL(g,Z)). We will show that this is only a first case in a series: in fact, for every pair (S,r) when S is a finite group with less than g generators and r is a Q-irreducible representation of S, we associate an arithmetic group which is then shown to be a virtual quotient of M. The case when S is the trivial group gives the above Sp(2g,Z) (resp., GL(g,Z)) but many new quotients are obtained. For example it is used to show that M(2) (resp., Aut(F3)) is virtually mapped onto a non-abelian free group. Another application is an answer to a question of Kowalski: generic elements in the Torelli groups are hyperbolic and fully irreducible. Joint work with Fritz Gruenwald, Michael Larsen and Justin Malestein.